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A162211 Number of reduced words of length n in the Weyl group D_8.

%I A162211 #21 Mar 21 2025 18:00:42
%S A162211 1,8,35,112,293,664,1350,2520,4388,7208,11263,16848,24248,33712,45425,
%T A162211 59480,75853,94384,114766,136544,159125,181800,203777,224224,242318,
%U A162211 257296,268504,275440,277788,275440,268504,257296,242318,224224,203777,181800,159125
%N A162211 Number of reduced words of length n in the Weyl group D_8.
%D A162211 N. Bourbaki, Groupes et alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche IV.)
%D A162211 J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.
%H A162211 Andrew Howroyd, <a href="/A162211/b162211.txt">Table of n, a(n) for n = 0..56</a>
%H A162211 <a href="/index/Gre#GROWTH">Index entries for growth series for groups</a>
%F A162211 The growth series for D_k is the polynomial f(k)*Prod_{i=1..k-1} f(2*i), where f(m) = (1-x^m)/(1-x) [Corrected by _N. J. A. Sloane_, Aug 07 2021]. This is a row of the triangle in A162206.
%p A162211 # Growth series for D_k, truncated to terms of order M. - _N. J. A. Sloane_, Aug 07 2021
%p A162211 f := proc(m::integer) (1-x^m)/(1-x) ; end proc:
%p A162211 g := proc(k,M) local a,i; global f;
%p A162211 a:=f(k)*mul(f(2*i),i=1..k-1);
%p A162211 seriestolist(series(a,x,M+1));
%p A162211 end proc;
%t A162211 n = 8;
%t A162211 x = y + y O[y]^(n^2);
%t A162211 (1-x^n) Product[1-x^(2k), {k, 1, n-1}]/(1-x)^n // CoefficientList[#, y]& (* _Jean-François Alcover_, Mar 25 2020, from A162206 *)
%Y A162211 Row 8 of A162206.
%Y A162211 Growth series for groups D_n, n = 3,...,50: A161435, A162207, A162208, A162209, A162210, A162211, A162212, A162248, A162288, A162297, A162300, A162301, A162321, A162327, A162328, A162346, A162347, A162359, A162360, A162364, A162365, A162366, A162367, A162368, A162369, A162370, A162376, A162377, A162378, A162379, A162380, A162381, A162384, A162388, A162389, A162392, A162399, A162402, A162403, A162411, A162412, A162413, A162418, A162452, A162456, A162461, A162469, A162492.
%K A162211 nonn,fini,full
%O A162211 0,2
%A A162211 _John Cannon_ and _N. J. A. Sloane_, Dec 01 2009